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1470 Ghaffari Materie.Pdf (6.362Mb) Wind and wave-induced currents over sloping bottom topography, with application to the Caspian Sea Thesis for the degree of Philosophiae Doctor Peygham Ghaffari Faculty of Mathematics and Natural Sciences University of Oslo Norway March 11, 2014 © Peygham Ghaffari, 2014 Series of dissertations submitted to the Faculty of Mathematics and Natural Sciences, University of Oslo No. 1470 ISSN 1501-7710 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission. Cover: Inger Sandved Anfinsen. Printed in Norway: AIT Oslo AS. Produced in co-operation with Akademika Publishing. The thesis is produced by Akademika Publishing merely in connection with the thesis defence. Kindly direct all inquiries regarding the thesis to the copyright holder or the unit which grants the doctorate. iii Acknowledgement Firstly, I am very grateful to my principal supervisor, Prof. Jan Erik H. Weber. Jan has always provided support, guidance and enthusiasm for this PhD. I appreciate his patience discussing, proof-reading and (most of all) for always looking for theoretical explanations. Thanks for being always helpful and supportive not only about scientific issues, but also with personal matters. Secondly, I would like to thank my second supervisor Prof. Joseph H. LaCasce, as well as Dr. Pål Erik Isachsen, for their valuable contributions. Many thanks to all friends and colleagues at the MetOs section for a great working environment. I wish to thank the Norwegian Quota Scheme for partial financial support during these years. And last but not least, I would like to thank my family Mahnaz and Marina for their patience and being always supportive. January, 2014 List of publications The present thesis is entitled "Wind and wave-induced currents over sloping bottom topography, with application to the Caspian Sea", and is based on the following papers: I- Ghaffari, P., Isachsen, P. E., and LaCasce, J. H.: 2013, Topographic effects on current variability in the Caspian Sea, J. Geophys. Res., 118, pp 1-10. II- Weber, J. E. H. and Ghaffari, P.: 2009, Mass transport in the Stokes edge wave, J. Mar. Res., 67, 213-224. II- Ghaffari, P. and Weber, J. E. H.: 2014, Mass transport in the Stokes edge wave for constant arbitrary bottom slope in a rotating ocean, J. Phys. Oceanogr., doi:10.1175/JPO-D-13- 0171.1, in press. IV- Weber, J. E. H. and Ghaffari, P.: 2014, Mass transport in internal coastal Kelvin waves, Europ. J. Mech. B/Fluids, in press. Contents 1 Introduction 1 1.1 Thesis objectives ................................. 1 2 Theory 3 2.1 On current variability over closed PV contours .................. 3 2.1.1 Barotropic model ............................. 3 2.1.2 Equivalent Barotropic model ....................... 5 2.2 Wave-induced flow ................................ 7 2.3 On the Stokes edge wave ............................. 8 2.3.1 Mass transport in the Stokes edge wave in a rotating ocean ....... 9 2.3.2 Mass transport in the Stokes edge wave in a non-rotating ocean .... 12 2.4 On internal waves ................................. 13 2.4.1 Mass transport in internal coastal Kelvin waves ............. 14 3 Observations 16 3.1 The Caspian Sea ................................. 16 3.2 Field measurements ................................ 18 3.2.1 Study area ................................. 18 3.2.2 Satellite observations ........................... 18 3.2.3 Current meter observations ........................ 18 3.2.4 Hydrographic observations ........................ 19 4 Results 22 4.1 Topographic Effects on the Large-scale Flow Field in the Caspian Sea ..... 22 4.2 Mass transport in the Stokes edge wave for constant arbitrary bottom slope . 26 4.3 Mass transport in internal coastal Kelvin waves ................. 28 5 Concluding remarks 31 Bibliography 31 Chapter 1 Introduction In the ocean basins topography and coasts may act to trap mechanical energy. This energy may be in the form of wind-driven currents along bottom contours or gravity waves along coasts or sloping beaches. Since waves do posses mean momentum when averaged over the wave cycle, i.e. the Stokes drift (Stokes, 1847), they induce a net transport of water particles in the same way as the more traditional wind-driven currents. Since these currents are trapped in regions close to the coast, or over bottom slopes in the coastal region, they may directly affect human activity and the coastal population through their ability to transport biological material, pollutants and sediments in suspension along the shore. 1.1 Thesis objectives The first objective of the work presented in this thesis is to provide better understanding of the topographic effect on current variability. This is done by exploring specific topics where the effect of topography may alter the currents. In wind-induced current case, the dynamics of flows with closed potential vorticity (PV) contours is examined. Topography generally distorts the geostrophic currents. If the topography is steep enough, it dominates the stationary part of the potential vorticity, even causing closed PV contours. Therefore, mean flow can be excited and exist on the closed contours (without encountering with boundaries) by wind-forcing. At ocean depths that are intersected by topography, currents steer around major topographic features. In addition, particularly at high latitudes, where the ocean is weakly stratified (Isachsen et al., 2003), geophysical flows tend to be vertically coherent (or barotropic) due to the rotation of the Earth. As a result currents near the ocean surface align in roughly the same direction as deep ocean currents, and consequently often follow contours of constant depth, e.g. see (Gille et al., 2004). Most major currents respond to sea floor topography. In the present study, we investigated to what degree this mechanism works for the Caspian Sea which is a low latitude water body. The second objective is to estimate the mean currents induced by trapped gravity waves. For that purpose, we investigated the Lagrangian mass transport in two different types of the gravity waves; the Stokes edge waves over sloping topography in a non-rotating and rotating ocean, and the internal coastal Kelvin wave. In practice, these waves are trapped due to a sloping bottom and the earth’s rotation. In particular, the Lagrangian mass transport is obtained from the ver- 2 Introduction tically integrated equation of momentum and mass, correct to second order in wave steepness. The Lagrangian current is composed of a non-linear Stokes drift (inherent in the waves) and a mean vertically-averaged Eulerian current. When waves propagate along topography (along the coast), the effect of viscosity leads to a wave attenuation, and hence a radiation stress com- ponent. The Eulerian current arises as a balance between the radiation stress, bottom friction, and the Coriolis force in a rotating ocean. The associated Stokes drift and Eulerian currents are investigated for the aforementioned waves. Additionally, their contributions to the mean Lagrangian drift are examined. We wish to explore how the contributions of the Stokes drift and Eulerian current varies with factors such as slope angle, wave amplitude, wave length and the earth’s rotation. In order to address these objectives four papers are included in this thesis. Objective number 1 is treated in paper I (Ghaffari et al., 2013), objective number 2 in papers II (Weber and Ghaffari, 2009), III (Ghaffari and Weber, 2014), and IV (Weber and Ghaffari, 2014). The remaining part of this thesis is organized as follows: chapter 2 presents a brief theory, and chapter 3 provides a description of the study area. Results are presented and discussed in chapter 4. This is then followed by prints of the papers. Chapter 2 Theory This section provides a short theoretical background of the wind-induced current variability on closed PV contours. Additionally, some theory and explanation of the concept of wave- induced currents, i.e. Stokes drift and Eulerian current are given. The section also presents a brief overview of the Stokes edge wave, the internal coastal Kelvin wave, and the associated wave-induced drifts. 2.1 On current variability over closed PV contours 2.1.1 Barotropic model Most major currents respond to sea floor topography. For instance, topography seems to steer the Antarctic Circumpolar Current (ACC) on certain part of its path (Gordon et al., 1978). Also Gulf Stream and the Kuroshio Extension all steer around ridges and seamounts. It is the conservation of potential vorticity which confines a water parcel to the PV contours. Hence, the conservation equation for PV provides an equally powerful and simple tool for discussion of the mean oceanic circulation on climatic scales (time scale of several months, length scale of about 500 km (Hasselmann, 1982)). In fact, strong topography may dominate the stationary part of the PV, even causing closed PV contours. The dynamics of the closed PV contours is different from those with blocked contours. With closed contours it is possible to obtain steady flows parallel to the PV contours. Such mean flow, which is not present with blocked contours, can be excited by wind forcing (Hasselmann, 1982; Kamenkovich, 1962; Greenspan, 1990). The mass transport, the depth integrated horizontal velocity, can be modeled using an integral equation derived from shallow water equation. In the linear limit, the model assumes a balance between divergence in the surface Ekman layer and convergence in the bottom Ekman layer; the latter is achieved via a flow parallel to the PV contours (Fig. 2.1). The flow strength can be predicted, if one knows the wind stress and topography. The theory is described by Isachsen et al. (2003). Consider a homogeneous ocean with a rigid lid and varying topography. Ignoring non-linearity and exploiting the Boussinesq approx- imation, the depth-integrated, horizontal momentum equation takes the form ∂u 1 τs + fk × u = −g∇η + − τb , (2.1) ∂t H ρ0 4 Theory Surface H Bottom Figure 2.1: Schematic of barotropic model spin-up.
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